How can we find the pole of square root function?

Question

What is the pole of the following function $$f(z)=\frac{\pi}{1+\sqrt{z}}, \quad z\in C$$

Answer 1

$$1+\sqrt z=0\implies z=(-1)^2=1.$$

But with the usual convention, $\sqrt1+1\ne0$, so there is no pole.

Answer 2

There is no "pole", but there is a "branch point" at $z=0$. Looking at the monodromy of the function around $z=0$, we see that the values change as $\sqrt{z}$ changes by $-1$. It doesn't "blow up" at $z=0$...